This talk is concerned with computational schemes for wall-bounded fluid flows. Functional integral representations for incompressible wall-bounded flows are established by using the reflection principle and a perturbation technique. These representations are expressed (implicitly) in terms of stochastic differential equations of McKean-Vlasov type, so that an exact random vortex method for a wall-bounded viscous flow is obtained. Numerical schemes are proposed and a numerical experiment is carried out for demonstrating the motion of a viscous flow within a thin layer next to the fluid boundary.

The talk is based on

[1] Zhongmin Qian, Youchun Qiu, Liang Zhao, Jiang-Lun Wu: Monte-Carlo simulations for wall-bounded fluid flows via random vortex method, arXiv:2208.13233

(https://arxiv.org/pdf/2208.13233.pdf).

[2] V. Cherepanov, S.W. Ertel, Z. Qian, J.-L. Wu: Random vortex dynamics and Monte-Carlo simulations for wall-bounded viscous flows, arXiv:2403.15549

(https://arxiv.org/pdf/2403.15549.pdf).